It is worth remarking that \(B^{1/3}_{3,q}\)  for \(1\le q<\infty \)  are included in \(B^{1/3}_{3,c(\mathbb {N})}\) , which is called as the Onsager-critical space.
In this direction, the recent progress can be found in
[1]}.
The proof of the other part of
Onsager's conjecture
in three-dimensional space is given by Isett [2]}, which adopted the approach developed
by De Lellis and Szekelyhidi [3]}, [4]}, [5]}, [6]}.
